Both functors \(|S_e \underline{U}_{(-)}|\)  and \(\mathrm {D}\underline{U}_{(-)}\)  take values in cofibrant topological spaces.
This has two implications:
first, by Lemma REF  and by the homotopical properties of the bar construction [1]} the top morphism in diagram (REF ) is a weak equivalence.
Second, both bar constructions model the homotopy colimits in \({\Delta {T}\mathrm {op}}\)  of the diagrams \(|S_e \underline{U}_{(-)}| \colon sA^\mathrm {op}\rightarrow {\Delta {T}\mathrm {op}}\)  and \(\mathrm {D}\colon sA^\mathrm {op}\rightarrow {\Delta {T}\mathrm {op}}\) , respectively.
